An Optimal Algorithm for solving the Dynamic Lot-Sizing Model with Learning and Forgetting in Setups and Production

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Int. J. Production Economics 95 (2005) 179–193

An optimal algorithm for solving the dynamic lot-sizing model with learning and forgetting in setups and production

Huan Neng Chiu

^a

, Hsin Min Chen

^b,

*

aDepartment of Industrial Management, National Taiwan University of Science and Technology, 43, Keelung Road, Section 4, Taipei 106, Taiwan

bDepartment of Business Management, National United University, No.1, Lien-Kung, Kung-Ching Li, Miaoli 360, Taiwan

Received 11 July 2001; accepted 2 December 2003

Abstract

This paper studies the problem ofincorporating both learning and forgetting in setups and production into the dynamic lot-sizing model to obtain an optimal production policy, including the optimal number ofproduction runs and the optimal production quantities during the finite period planning horizon. Since the unit production cost is variable due to the effects of learning and forgetting, the first-in-first-out (FIFO) inventory costing method is used in our model.

After deriving the relevant cost functions, we develop the multi-dimensional forward dynamic programming (MDFDP) algorithm based on two important properties that can be proved to be able to reduce the computational complexity. A numerical example is illustrated and solved using our refined MDFDP algorithm. The results from our computational experiment show that the optimal number ofproduction runs decreases with the increase ofthe learning or forgetting rates, while the optimal total cost increases with the increase ofone ofthe above four rates. Production learning has the greatest influence on the optimal total cost among the four parameters. The interactive effects of five demand patterns and nine relationships generated by the four rates on the optimal number of production runs and the optimal total cost are also examined.

Keywords: Learning and forgetting effects; Dynamic lot-sizing model; FIFO inventory costing method

1. Introduction

Owing to the increasing emphasis on time-based competition, the importance oflearning and forgetting effects on manufacturing has been widely recognized. Both effects on the continuous review system with a constant demand rate have been studied byKeachie and Fontana (1966),Spradlin and Pierce (1967),Adler and Nanda (1974),Carlson (1975),Sule (1978, 1981),Axs.ater and Elmaghraby (1981),Elmaghraby (1990), and Jaber and Bonney (1997a, 1998, 2001). The above studies only considered learning and forgetting effects on production. Another study conducted by Li and Cheng (1994) was more general in that the economic production quantity (EPQ) model involved learning in setups and both learning and forgetting in production.Jaber and Bonney (1999)surveyed the above models and suggested possible extensions to the

*Corresponding author. Tel.: +886-37-381590; fax: 886-37-332396.

E-mail address:[email protected] (H.M. Chen).

doi:10.1016/j.ijpe.2003.12.001

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lot size problem in which both learning and forgetting are incorporated into both setups and production.

They also suggested that their earlier work may be extended to the model ofWagner and Whitin by including a ﬁnite planning horizon with zero inventories at the beginning ofthe initial cycle and the end of the last cycle. However, few papers have incorporated both effects into the dynamic lot-sizing problems with discrete time-varying demand.Chand and Sethi (1990)considered the dynamic lot-sizing problem in a pure setup learning environment in which only setup costs were susceptible to improvements. They developed a forward dynamic programming algorithm, which can be used on a rolling horizon basis, for inﬁnite horizon problems.Tzur (1996)extended Chand and Sethi’s work to a more general model, which allows the total setup cost to be a general nondecreasing (but not necessarily concave) function of the number ofsetups. Recently,Chiu (1997)incorporated learning and forgetting effects on production into the dynamic lot-sizing model. Furthermore, he also extended the optimalWagner and Whitin (1958)algorithm and three existing heuristic models.

Unlike previous works, this paper studies the problem ofincorporating both learning and forgetting in setups and production into the dynamic lot-sizing model to obtain an optimal production policy, including the number ofproduction runs, lot sizes, and time points for starting setups and production. Since the period-demand and ﬁnite periods ofthe planning horizon are assumed in this paper, but setups and production times are scheduled continuously, the proposed model is virtually a mix ofdiscrete and continuous models. As far as we know, few papers have studied this model.

The setup time and unit production time are assumed to have learning phenomena, and are represented as power functions of the cumulative number of repetitions. The forgetting effect is mainly caused by a break between two consecutive production runs and leads to retrogression in learning. Besides the quantity produced to date and the length ofthe interruption, other factors such as the availability ofthe same personnel, tooling, and methods that have a direct effect on the degree of human forgetting were also considered in Anderlohr (1969)and Cochran (1973). Globerson et al. (1989) showed that the degree of forgetting is a function of the break length and the level of experience gained prior to the break in a laboratory experiment. In fact, a variety of factors inﬂuence the forgetting effect like the break length, previous experience, job complexity, the work engaged in during the interruption period, the cycle time of the task, the relearning curve, and a single relearning observation (e.g., repair or maintenance) (Dar-El, 2000, pp. 83–92).Jaber and Bonney (1996)proposed a mathematical model in which the forgetting slope is dependent on three factors (i.e., the equivalent accumulated output of continuous production by the point ofinterruption, the minimum break under total forgetting, and the learning slope). They (Jaber and Bonney, 1997b) also compared their model with two existing models. Their model is more realistic, and their predicted time was very close to the experimental data provided by Globerson et al. (1989). For simplicity, we assumed ﬁxed forgetting rates in setups and production, as adopted byLi and Cheng (1994), to make our proposed multi-dimensional forward dynamic programming (MDFDP) algorithm more tractable. Since the production cost ofeach unit is not identical due to learning and forgetting, the FIFO inventory costing method is used in this paper.

In the next section, the notations used throughout this paper are deﬁned, and basic assumptions are given. Section 3 then presents a general description of the model and formulates relevant cost functions for each production run. Subsequently, Section 4 develops the reﬁned MDFDP algorithm by applying two important properties, and an example is also provided. An experiment conducted to analyze the effects of relevant parameters on the optimal solution is discussed in Section 5. Finally, Section 6 concludes the paper with a briefsummary ofthe results.

2. Notations and assumptions

The following notations will be used throughout this study:

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Parameters

N length ofthe planning horizon expressed in periods di demand in a given period i; d1> 0 and diX0 rs ﬁxed learning rate in setups, 0orsp1

bs learning coefﬁcient associated with setups, bs¼ log r_s=log 2 fs ﬁxed forgetting rate in setups, 0pfsp1

rp ﬁxed learning rate in production, 0orpp1

bp learning coefﬁcient associated with production, bp¼ log r_p=log 2 f_p ﬁxed forgetting rate in production, 0pfpp1

y ﬁxed production capacity per period (in man-periods) C_o direct labor cost per man-period

C_m direct material cost and overhead per unit C_h ﬁxed carrying cost rate per period

Decision variables

n total number ofproduction runs planned for the entire planning horizon qj number ofunits produced in the jth production run.

Intermediate variables

i period count that denotes the time interval between the time points of i 1 and i; i ¼ 1; 2; y; N j production run count, j ¼ 1; 2; y; and jpN

Dði; mÞ cumulative units ofdemand from a speciﬁc period i to period m: That is, Dði; mÞ ¼ d_iþ d_iþ1þ

?þ d_m1þ d_m¼Pm

a¼1da and ipmpN

I ð i Þ inventory at the end ofperiod i after the demand d_i is satisﬁed, I ð i ÞX0

Q_j cumulative units produced from the ﬁrst production run to the jth production run. That is, Qj¼ q1þ q2þ ? þ qj and Q0¼ 0

Sj time (in man-periods) required to set up the jth production run

tj; x time (in man-periods) required to produce the xth cumulative unit ofthe jth production run, where 1pxpqj

Pj production time in the jth production run, Pj¼P^qj x¼1tj; x

Aj time point at which setup ofthe jth production run begins (seeFig. 1) Bj time point at which production in the jth production run begins

Mj number ofperiods whose demand is satisﬁed during the production phase (Phase I) in the jth production run

K_j number ofperiods whose demand is satisﬁed during the non-production phase (Phase II) in the jth production run

SC_j the setup cost ofthe jth production run

PC_j the production cost, including the direct labor cost, direct material cost and overhead, for the jth production run

WC_j the inventory carrying cost incurred during the production phase in the jth production run

HCj the inventory carrying cost incurred during the non-production phase in the jth production run

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The objective ofthis paper is to obtain the optimal solution for the above deﬁned decision variables that minimizes the total cost during the planning horizon i.e., minimize

Xⁿ

j¼1

ðSC_jþ PC_jþ WC_jþ HC_jÞ:

Inventory level

i i+1 i+K_j−1 i+K_j i+K_j+M_j

i i+1 i+K_j−1 i+K_j+M_j

phase I phase II

Bj B_j+P_j

−1 i

−1 i 0

0 ) 1 (i− I

) , ( ii U

) 1 , (ii+ U qj

Cumulative units

The length of production phase The length of non-production phase

Kj i+ Aj

S j

The jth production run

di

+1 di

−1 +Kj

di

Kj

di₊

+1 di

DELIVERED Kj

di₊

−1 +Kj

di

) 1 ( −

−Ii d_i

Fig. 1. Inventory levels and cumulative production units ofthe jth production run.

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The following assumptions are made herein:

(1) The single-stage manufacturing system considers only one product, and the product is not subject to deterioration, obsolescence, or perishability during the ﬁnite planning horizon.

(2) The demand in the form of d1; d2; y; dN is known but varies from one period to another. The demand for each period is scheduled to be delivered (i.e., to be satisﬁed) at the end of that period, and each period has the same length oftime.

(3) The beginning inventory in period 1 and the ending inventory in period N both equal zero (i.e., I ð0Þ ¼ I ðN Þ ¼ 0). No shortages or stockouts are permitted during the planning horizon. The production capacity per period, y; can satisfy the period demand. A mathematical expression for the production capacity constraint is S_jþPdi

x¼1t_{j; x}py man-periods, where the jth production run during the planning horizon is performed in period i: Without loss ofthe generality, we assume that y ¼ 1 in this paper.

(4) To achieve the objectives oflower inventory and on-time delivery, the start times ofsetup and production in a production period are delayed as long as possible without incurring shortages.

Production starts immediately when setup is ﬁnished. A setup is not necessarily incurred in every production period, but only occurs after non-production (idle time).

(5) The FIFO rule is used to govern delivery units ofthe product produced.

(6) Both the setup time and unit production time decrease as a result oflearning. A ﬁxed fraction ofthe total setup learning is lost (i.e., forgotten or retrogressed) due to a manufacturing interruption between two consecutive setups. Forgetting is similarly applied to production. The two forgetting rates (i.e., fs

and fp) have been defined previously. This forgetting assumption in production was used byLi and Cheng (1994). The time required to set up the first production run, denoted by S1; and the time required to produce the first unit ofthe first production run, t1;1; are both known.

(7) Cost parameters Co; Cmand Chdo not change with time. The direct labor cost per period is constant since we assume that the same skilled workers perform the setup and production jobs. It is also assumed that the total overhead during the planning horizon can be estimated and allocated to the total production quantities ðQ_nÞ: Hence, the value of C_m(i.e., the sum ofthe unit direct material cost and unit overhead) is ﬁxed. Similarly, this ﬁxed value of C_hcan be easily estimated based on the current cost ofcapital.

(8) The carrying cost for a unit of the product is proportional to its production cost and is calculated based on the time length from its completion time to the time when it is delivered.

3. Model description

The learning functions without forgetting in setups and production are

Sj¼ S₁½ð j 1Þ þ 1^b^s ¼ S₁j^b^s ð1Þ

and

t_{j; x}¼ t_1;1ðQ_j1þ xÞ^b^p; ð2Þ

where 1pxpqj: From Assumption (6), the time required to set up the jth production run is

S_j¼ S₁½ð1 f_sÞð j 1Þ þ 1^b^s; ð3Þ

where 1 fs represents the retentive proportion ofthe total learning obtained in the previous j 1 setups.

Similarly, the production time required to produce the xth unit in the jth production run is

tj; x¼ t_1;1½ð1 f_pÞQ_j1þ x^b^p: ð4Þ

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Obviously, Sjin Eq. (3) equals S₁when j ¼ 1 (the ﬁrst production run), and tj; xin Eq. (4) equals t_1;1when j ¼ 1 and x ¼ 1: In addition, Eq. (4) implies that a fraction of the total learning deﬁned by Li and Cheng (1994, p. 121, Eq. (2)) is lost between production lots. Alternatively, ifwe make the remembered learning assumption under which the loss is related to the cumulative units remembered, then Eq. (4) becomes tj; x¼ t_1;1½Pj1

i¼1ð1 f_pÞ^jiqiþ x^b^p: As stated byLi and Cheng (1994), such an assumption would lead to a more complex model to which the dynamic programming approach could not be applied.

From Eq. (3) and Assumption (7), the setup cost ofthe jth production run is

SC_j¼ C_oS_j¼ C_oS₁½ð1 f_sÞð j 1Þ þ 1^b^s: ð5Þ

The production cost, including the direct labor cost, direct material cost and overhead, for the jth production run can be derived from Eq. (4) and Assumption (7). The result is given by

PCj¼ C_oPjþ C_mqj ¼ C_o X^q^j

x¼1

tj; xþ C_mqj

¼ C_ot_1;1 X^q^j

x¼1

½ð1 f_pÞQ_j1þ x^b^pþ C_mq_j: ð6Þ

As shown in Fig. 1, the jth production run is supposed to start production at time Bj in period i (i.e., i 1oBjX1). Because stockouts are not allowed, as described in Assumption (3), 0odi Iði 1Þp q_jpDð1; N Þ Qj1: Meanwhile, the time at which the jth production run begins to produce can be determined by

B_j¼ i ^dⁱ^Iði1ÞX

x¼1

t_{j; x}> i 1; ð7Þ

where i here represents the time length from the beginning ofperiod 1 to the end ofperiod i: The time at which setup ofthe jth production run begins is

Aj¼ B_j S_jXi 1: ð8Þ

The carrying cost for the units produced in the jth production run can be divided into two parts. One part ofthe carrying cost (see the left shaded area inFig. 1) is calculated in Phase I, while another part ofthe carrying cost (see the right shaded area inFig. 1) is computed in Phase II.

In Phase I, the number ofperiods in which each period-demand is satisﬁed by the quantity produced in the jth production run is

K_j¼ IB_jþ P_jm ði 1Þ; ð9Þ

where IBjþ Pjm denotes the largest integer no greater than Bjþ Pj: To simplify our presentation, we deﬁne that U ði; wÞ ¼ Dði; wÞ I ði 1Þ and ipwpN; given Iði 1Þ: The carrying cost based on every unit production cost in this phase (i.e., the time interval between Bjand Bjþ P_j) can then be derived as shown in Appendix A. The result is

WCj¼ C_h ^qX^j¹

x¼1

ðC_otj; xþ C_mÞ X^q^j

y¼xþ1

tj; y

!

" #

C_o ^{U ði;iÞ}X

x¼1

tj; xþ C_mU ði; iÞ

" #

ðB_jþ P_j iÞ (

^iþKX^j²

w¼i

Co

X

U ði;wþ1Þ

x¼U ði;wÞþ1

tj; xþ C_mdwþ1

!

½B_jþ P_j ðw þ 1Þ

)

: ð10Þ

Thus, the number ofperiods in which each period-demand is satisﬁed in Phase II is given by

Mj¼ maxfinteger g jD½1; ði 1Þ þ K_jþ gpQjg: ð11Þ

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The carrying cost in this phase (from time Bjþ Pj to time i þ Kjþ Mj), calculated in Appendix B, can be presented as

HC_j¼ C_h ^iþKX^j^þM^j¹

a¼iþKj

C_o ^{U ði;aÞ}X

x¼U ði;a1Þþ1

t_{j; x}þ C_md_a

!

½a ðB_jþ P_jÞ

8<

:

þ C_o X^q^j

x¼U ði;iþKjþMj1Þþ1

tj; xþ C_mðQ_j Dð1; i þ K_jþ M_j 1ÞÞ 2

4

3

5½i þ Kjþ M_j ðB_jþ P_jÞ

9=

;: ð12Þ Consequently, the total cost ofthe jth production run, which starts production in period i; is

TCj¼ TCði; j; Q_j1; qjÞ

¼ SC_jþ PC_jþ WC_jþ HC_j; ð13Þ

where SCj; PCj; WCj; and HCj are given in Eqs. (5), (6), (10), and (12), respectively.

It should be noted here that the time length ofPhase II in the jth production run should be long enough so that the ð j þ 1Þth production run can be setup and satisﬁes the net demand (i.e., d_iþK_jþMj2I ði þ K_jþ M_j 1Þ) at the end ofperiod i þ K_jþ M_j: That is,

S_jþ1þ^d^iþK^j^þM^j^IðiþKX ^j^þM^j^1Þ

x¼1

t_jþ1;xpi þ Kjþ M_j ðB_jþ P_jÞ; ð14Þ

where I ði þ Kjþ M_j 1Þ ¼ Q_jP^iþKjþMj1

a¼1 da:

The mathematical model ofthis research problem is as follows:

Minimize Xⁿ

j¼1

ðSC_jþ PC_jþ WC_jþ HC_jÞ

subject to S_jþX^dⁱ

x¼1

t_{j; x}p1 for 1pjpipN;

0pIð i ÞpDð1; N Þ Dð 1; i Þ; for i ¼ 1; 2; y; N;

d_ipIði 1Þ þ qj; 0oqjpDð1; N Þ Qj1; I ð0Þ ¼ 0;

1pnpN:

The ﬁrst inequality, S_jþPdi

x¼1t_{j; x}p1; expresses the capacity constraint, as described in Assumption (3).

The second inequality, 0pIð i ÞpDð1; N Þ Dð1; iÞ; implies that IðN Þ ¼ 0: The third inequality, dip I ði 1Þ þ q_j; represents the assumption under which no shortages are permitted. The last inequality, 0oqjpDð1; N Þ Qj1; constrains the number ofunits produced in the jth production run that does not exceed an upper limit under the assumption that I ðN Þ ¼ 0: The upper limit is determined by subtracting Qj1from the total demand during the planning horizon.

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4. The optimal multi-dimensional forward dynamic programming algorithm

Since the total cost ofthe jth production run, as shown in Eq. (13), depends on i; j; Qj1; and qj; and the mathematical model mentioned in Section 3 cannot be solved directly, the proposed MDFDP algorithm reﬁned by applying Properties 1 and 2 can be used to solve the optimal values of q1; q2; y; qn

and n:

Property 1. The optimal solution does not include a production run started in a period in which the beginning inventory is large or equal to the demand of that period.

Proof. Suppose the jth production run is performed and produces q units in period i; where I ði 1ÞXd_i and qpqj: According to Assumption (3), postponement ofthe jth production run to the period i þ 1 is beneﬁcial since the savings obtained in the carrying cost is

Ch Cot1;1

X^q

x¼1

½ð1 f_pÞQ_j1þ x^b^p þ C_mq

( )

> 0: &

First, let Lð j Þ be the period in which the jth production run is set up and begins to perform production.

The total cost function is deﬁned as follows:

F ½Lð j Þ; j; Q_j = the minimum total cost from the ﬁrst production run to the jth production run, given that the jth production run is set up and begins production in period Lð j Þ; where jpLð j ÞpN; that the cumulative production quantities is Qj; and that Qj is sufﬁcient to satisfy the demand from period 1 to period Lð j Þ:

Second, the recurrence relation is F ½Lð j Þ; j; Qj

¼ minfTC½Lð j Þ; j; Q_j q_j; q_j þ F ½Lð j 1Þ; j 1; Q_j q_jjLð j 1ÞoLð j ÞpN;

0pIðLð j ÞÞpDð1; N Þ Dð1; Lð j ÞÞ; dLð j ÞpQj Dð1; Lð j Þ 1Þ; and q_jpDð1; N Þ Qj1g: ð15Þ Third, the boundary conditions are F ð0; 0; 0Þ ¼ 0; F ði; 0; 0Þ-N for iX1; Fð0; j; 0Þ-N for jX1; and F ð0; 0; QjÞ-N for QjX1: Finally, the optimal solution is

F _n ¼ F ½LðnÞ; n; Dð1; N Þ ¼ minfFng; where n ¼ 1; 2; y; N and

F_n¼ F ½LðnÞ; n; Dð1; N Þ ¼ minfTC½LðnÞ; n; Dð1; N Þ q_n; q_n þ F ½Lðn 1Þ; n 1; Dð1; N Þ q_ng:

As a result, the optimal values of q_j for j ¼ 1; 2; y; n can be obtained by using the backtracking process.

Here, the computational complexity ofEq. (15) is OðNðDð1; N ÞÞ²Þ: Further, improvement ofthe computational efﬁciency can be achieved by means of the following property:

Property 2. If the production learning rate is fixed and the unit inventory carrying cost per period for the product is proportional to the unit production cost, which is variable due to learning, then the zero inventory property holds.

Proof. Since the unit production time decreases with the increase in the number ofunits produced as a result ofthe ﬁxed production learning rate, both the unit production cost and the unit inventory carrying cost per period are nonincreasing (concave) functions. From Taha (1997, p. 462), it is easy to show that I ði 1Þqj¼ 0 for all i (where Iði 1Þ is the beginning inventory in period i; and j is the count ofthe next production run occurring in period i). &

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From Property 2, Eq. (15) can be simpliﬁed to obtain

F ði; j; QjÞ ¼ min TC½i; j; Dð1; i 1Þ; q_j þ F ½Lð j 1Þ; j 1; Dð1; i 1ÞjLð j 1ÞpipN;

(

I ði 1Þ ¼ 0; d_i> 0; and q_j¼X^l

a¼i

d_a; where l ¼ i; i þ 1; y; N )

: ð16Þ

The computational complexity ofthe proposed MDFDP algorithm can be reduced from OðNðDð1; N ÞÞ²Þ to OðN³Þ since NpDð1; N Þ:

Example. A producer of industrial vehicles carried out pilot production to satisfy orders for a new type of straddle carrier. The ﬁnished carriers were periodically delivered by train, and the production manager was confronted with the following ordering situation for the ﬁrst 6 periods:

Period i 1 2 3 4 5 6

Demand di 6 9 11 5 3 15

Given S₁ ¼ 0:25; t_1;1¼ 0:05; C_o¼ 1; 000; C_m¼ 500; and C_h¼ 0:05; suppose that r_s¼ 0:80; f_s¼ 0:60; r_p¼ 0:90; and f_p¼ 0:40: Using the reﬁned MDFDP algorithm, the best production policies for n ¼ 1; 2; y; and 6 were those summarized in Table 1. As n increases, the inventory carrying cost reduces but setups and production costs increase. Therefore, the production policy with an adequate value of n is advantageous. In this example, zero inventories are encountered 8.25% (i.e., 0:4950=6 100%Þ and 20.77%

Table 1

Results for the numerical example

n j Aj Sj Bj Qj Pj SCj PCj WCj HCj TCj Fn F _n

1 1 0.4950 0.2500 0.7450 49 1.5795 250.00 26079.50 643.63 2088.44 29061.60 29061.60 2 1 0.4950 0.2500 0.7450 31 1.0660 250.00 16566.00 265.94 683.14 17765.10

2 4.6810 0.2243 4.9053 18 0.5437 224.34 9543.72 84.69 218.99 10071.70 27836.80 3 1 0.4950 0.2500 0.7450 15 0.5692 250.00 8069.19 49.16 165.09 8533.44

2 2.4098 0.2243 2.6341 19 0.6121 224.34 10112.10 70.49 239.59 10646.50

3 5.3412 0.2069 5.5481 15 0.4519 206.90 7951.91 82.70 0.00 8241.51 27421.40 27421.40 4 1 0.4950 0.2500 0.7450 15 0.5692 250.00 8069.19 49.16 165.09 8533.44

2 2.4098 0.2243 2.6341 11 0.3659 224.34 5865.90 47.82 0.00 6138.06 3 3.6327 0.2069 3.8396 8 0.2538 206.90 4253.79 10.94 72.23 4543.86

4 5.3542 0.1939 5.5481 15 0.4519 193.96 7951.91 82.70 0.00 8228.57 27443.90 5 1 0.4950 0.2500 0.7450 6 0.2550 250.00 3255.04 16.31 0.00 3521.34

2 1.4484 0.2243 1.6727 9 0.3273 224.34 4827.29 34.08 0.00 5085.71 3 2.4272 0.2069 2.6341 11 0.3659 206.90 5865.90 47.82 0.00 6120.62 4 3.6457 0.1939 3.8396 8 0.2538 193.96 4253.79 10.94 72.23 4530.91

5 5.3643 0.1838 5.5481 15 0.4519 183.80 7951.91 82.70 0.00 8218.42 27477.00 6 1 0.4950 0.2500 0.7450 6 0.2550 250.00 3255.04 16.31 0.00 3521.34

2 1.4484 0.2243 1.6727 9 0.3273 224.34 4827.29 34.08 0.00 5085.71 3 2.4272 0.2069 2.6341 11 0.3659 206.90 5865.90 47.82 0.00 6120.62 4 3.6457 0.1939 3.8396 5 0.1604 193.96 2660.40 8.47 0.00 2862.82 5 4.7215 0.1838 4.9053 3 0.0947 183.80 1594.72 2.51 0.00 1781.03

6 5.3726 0.1755 5.5481 15 0.4519 175.53 7951.91 82.70 0.00 8210.14 27581.70

Note: For example, n ¼ 3; the time point for starting setup for the ﬁrst production run (i.e., j ¼ 1) is 0.4950. The duration ofthe setup is 0.2500; consequently, setup ends at time 0.7450 (=0.4950+0.2500).

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(i.e., ð0:4950 þ 0:4098 þ 0:3412Þ=6 100%Þ ofthe time for n=1 and 3, respectively. When n increases from 1 to 3, setups and production costs increase by 484.84 and the inventory carrying cost reduces by 2,125.04.

Hence, the total cost reduces by 1,640.20. However, as n increases from 3 to 4, setups and production costs increase by 201.55 but the inventory carrying cost only reduces by 179.05. The total cost increases by 22.50.

Similarly, the total costs are increasing as n increases from 4 to 6. As a result, the optimal production policy was based on three runs (i.e., n ¼ 3) during the 6-period planning horizon (i.e., N ¼ 6). The three runs were set up at times 0.4950 (see the note in Table 1), 2.4098, and 5.3412. Each production run was started immediately when the corresponding setup was ﬁnished; i.e., the three runs were started at 0.7450 (also see note inTable 1), 2.6341, and 5.5481. In fact, the three runs produced 15 units, 19 units, and 15 units in period 1, 3, and 6, respectively. The minimum total cost was 27,421.40.

5. Computational experience

We conducted an experiment to explore the effects of learning, forgetting, and the demand pattern on the total cost and the number ofproduction runs. The proposed MDFDP algorithm was programmed in Visual C++ 6.0 and run on a PC with a Pentium III 600. A series ofproblems generated from Dð1; N Þ ¼ 60 were tested. For each test problem, the ﬁxed parameters, including N; S₁; t_1;1; C_o; C_m; and C_h; were assigned the same values presented in the previous section. The various values for each of the other parameters were r_s¼ 0:6; 0.8, and 1.0; f_s¼ 0:0; 0.5, and 1.0; r_p¼ 0:6; 0.8, and 1.0; and f_p¼ 0:0; 0.5, and 1.0. In addition, the ﬁve types ofdemand patterns were chosen as follows:

Type I. Demand concentrated in the early and late periods: 15, 10, 5, 5, 10, 15.

Type II. Demand concentrated in the middle periods: 5, 10, 15, 15, 10, 5.

Type III. Equal demand in all periods: 10, 10, 10, 10, 10, 10.

Type IV. Gradually descending demand: 15, 15, 10, 10, 5, 5.

Type V. Gradually ascending demand: 5, 5, 10, 10, 15, 15.

A total of405 (i.e., 3 3 3 3 5) test problems were generated.Tables 2, 3 and 4present the results obtained by using the proposed MDFDP algorithm. They are explained in the following:

(1) Table 2shows that the average optimal number ofproduction runs decrease slightly with the increase ofthe values ofrs; fs; rp; or fp:

Table 2

Average optimal number ofproduction runs obtained by using the proposed algorithm

Parameters rp¼ 0:6 rp¼ 0:8 rp¼ 1:0 Row

average Overall average fp¼ 0:0 fp¼ 0:5 fp¼ 1:0 fp¼ 0:0 fp¼ 0:5 fp¼ 1:0 fp¼ 0:0 fp¼ 0:05 fp¼ 1:0

rs¼ 0:6 fs¼ 0:0 5.00 5.00 4.00 5.00 5.00 3.77 4.15 4.15 4.15 4.47

fs¼ 0:5 5.00 5.00 3.38 5.00 4.85 3.15 3.54 3.54 3.54 4.11 3.94

fs¼ 1:0 3.54 3.54 3.00 3.54 3.54 3.00 3.00 3.00 3.00 3.24

rs¼ 0:8 fs¼ 0:0 4.93 4.93 3.93 4.93 4.93 3.67 4.00 4.00 4.00 4.37

f_s¼ 0:5 4.93 4.93 3.40 4.93 4.80 3.13 3.47 3.47 3.47 4.06 3.90

f_s¼ 1:0 3.60 3.60 3.00 3.60 3.60 3.00 3.00 3.00 3.00 3.27

r_s¼ 1:0 f_s¼ 0:0 4.93 4.93 3.93 4.93 4.93 3.67 4.00 4.00 4.00 4.37

f_s¼ 0:5 4.93 4.93 3.40 4.93 4.80 3.13 3.47 3.47 3.47 4.06 3.90

f_s¼ 1:0 3.60 3.60 3.00 3.60 3.60 3.00 3.00 3.00 3.00 3.27

Column average 4.50 4.50 3.45 4.50 4.45 3.28 3.51 3.51 3.51

Overall average 4.15 4.08 3.51

(11)

(2) Table 3indicates that the average optimal total cost increased with the increase ofthe values ofrs; fs; rp; or fp: It is apparent that the increase ofthe value ofeach parameter led directly to a higher setup or unit production cost. In addition, production learning had the greatest inﬂuence on total cost among the four parameters.

(3) The observation inTable 2implies that the effects of r_pon the number ofproduction runs are more inﬂuential than that of r_s: For instance, as r_pand r_sdecrease from 1.0 to 0.6, variations in the optimal number ofproduction runs are 18.23% (i.e., ð4:15 3:51Þ=3:51 100%Þ and 1.03% (i.e., ð3:94 3:90Þ=3:90 100%Þ; respectively.

(4) InTable 3, it can be seen that the effect of r_pon the total cost is more signiﬁcant than that of r_son the total cost. For example, as r_p and r_s go from 1.0 to 0.6, variations of total cost are 7.55% (i.e., (34,447.2431,845.11)/34,447.24%Þ and 0.03% (i.e., (32,997.0932,987.81)/32,997.09100%Þ; respectively.

(5) Tables 2 and 3also further reveal the important result that the smaller the values of rsand rpare, the more inﬂuential fsand fpare. This result is consistent with the ﬁndings ofJaber and Bonney (1996)and

Table 3

Average optimal total cost obtained by using the proposed algorithm

Parameter rs¼ 0:6 rp¼ 0:8 rp¼ 1:0 Row

average

Overall average fp¼ 0:0 fp¼ 0:5 fp¼ 1:0 fp¼ 0:0 fp¼ 0:5 fp¼ 1:0 fp¼ 0:0 fp¼ 0:5 fp¼ 1:0

r_s¼ 0:6 f_s¼ 0:0 31477.39 31571.51 32091.80 32310.31 32462.36 32939.05 34355.82 34355.82 34355.82 32879.99

f_s¼ 0:5 31577.97 31672.09 32164.17 32410.87 32562.36 32998.28 34436.78 34436.78 34436.78 32966.23 32987.81 f_s¼ 1:0 31823.78 31897.08 32268.05 32626.41 32750.39 33087.75 34533.80 34533.80 34533.80 33117.21

r_s¼ 0:8 f_s¼ 0:0 31500.68 31592.17 32106.30 32333.73 32482.82 32949.99 34371.25 34371.25 34371.25 32897.72

fs¼ 0:5 31594.29 31685.79 32173.69 32427.33 32575.93 33004.25 34444.34 34444.34 34444.34 32977.14 32997.09 fs¼ 1:0 31821.20 31893.18 32269.78 32626.10 32748.86 33087.16 34533.80 34533.80 34533.80 33116.41

rs¼ 1:0 fs¼ 0:0 31500.68 31592.17 32106.30 32333.73 32482.82 32949.99 34371.25 34371.25 34371.25 32897.72

fs¼ 0:5 31594.29 31685.79 32173.69 32427.33 32575.93 33004.25 34444.34 34444.34 34444.34 32977.14 32997.09 fs¼ 1:0 31821.20 31893.18 32269.78 32626.10 32748.86 33087.16 34533.80 34533.80 34533.80 33116.41

Column average 31634.61 31720.33 32180.40 32457.99 32598.93 33011.99 34447.24 34447.24 34447.24

Overall average 31845.11 32689.63 34447.24

Table 4

A summary ofthe average optimal number ofproduction runs and average optimal total cost Relationship of

parameters

Average optimal number ofproduction runs Average optimal total cost

Demand pattern Row

average

Demand pattern Row

average

I II III IV V I II III IV V

rsorp fso fp 4.44 4.33 4.67 4.33 4.33 4.42 33766.22 33768.52 33751.28 33763.71 33770.08 33763.96 fs¼ fp 4.11 4.11 4.33 4.11 4.00 4.13 33774.43 33775.39 33758.59 33773.88 33773.76 33771.21 fs> fp 3.67 3.67 3.67 3.44 3.67 3.62 33836.53 33834.70 33834.51 33837.32 33834.44 33835.50 rs¼ r_p fso fp 3.89 4.00 4.00 3.89 3.89 3.93 33031.52 33036.44 33058.59 33027.90 33036.67 33038.22 fs¼ f_p 4.11 4.00 4.33 4.11 4.11 4.13 32949.64 32952.58 32959.48 32941.31 32948.70 32950.34 fs> fp 4.00 4.00 3.67 3.56 4.00 3.85 32938.48 32935.37 32970.76 32941.11 32935.12 32944.17 rs> rp fso fp 3.67 3.56 3.33 3.22 3.56 3.47 32358.97 32368.51 32396.21 32328.56 32363.58 32363.17 fs¼ f_p 3.89 3.89 3.67 3.44 3.89 3.76 32206.89 32208.32 32251.80 32190.78 32205.21 32212.60 fs> fp 4.11 4.11 3.33 3.22 4.11 3.78 32081.36 32074.81 32154.47 32085.52 32077.00 32094.63 Column average 3.99 3.96 3.89 3.70 3.95 32993.78 32994.96 33015.08 32987.79 32993.84

(12)

Jaber and Kher (2002)in which the forgetting effects are dependent on the learning effects. InTable 2, as rs decreases from 1.0 to 0.6, the effect of fs on the optimal number ofproduction runs (shown in Row average of Table 2) increases from 33.64% (i.e., ð4:37 3:27Þ=3:27 100%Þ to 37.96% (i.e., ð4:47 3:24Þ=3:24 100%Þ: Similarly, as r_p decreases from 1.0 to 0.6, the effect of fpon the optimal number ofproduction runs (shown in Column average ofTable 2) increases from 0.00% (i.e., ð3:51 3:51Þ=3:51 100%Þ to 30.88% (i.e., ð4:50 3:45Þ=3:45 100%Þ: Results in Table 3 also show that effects of fs and fpon total cost increase as rs and rpdecrease from 1.0 to 0.6, respectively.

(6) The optimal number ofproduction runs and the optimal total cost were insensitive to the demand pattern, as shown inTable 4. Nine relationships among rs; fs; rp; and fpare shown inTable 4. It can be observed from the ﬁrst three relationships that when r_sorp; the optimal number ofproduction runs decreased as the forgetting rate in setups ðf_sÞ relative to the forgetting rate in production ðf_pÞ increased.

The main reason is that the smaller r_sand the larger f_sincurred a higher cost in setups. The next three relationships led to the same results. However, the last three relationships exhibited the opposite phenomenon since the effects on production surpassed those on setups.

6. Conclusions

This study has presented an effective approach to handling the complex dynamic lot-sizing model, in which learning and forgetting in setups and production are considered simultaneously. In fact, the proposed MDFDP model is a mix ofdiscrete and continuous ones. This inevitably causes intractability in obtaining the optimal solution, including the optimal number ofproduction runs and the optimal production quantities during the planning horizon. Fortunately, we have developed two important properties that have been proven able to reduce the computational complexity.

The results shown in our computational experience have indicated that the average optimal number of production runs decreases as one rate increases, and that the other three rates remain fixed. The average optimal total cost increases as one ofthe above four rates increases. Furthermore, production learning has the greatest influence on the optimal total cost, as compared with the other three parameters. The effects of production learning on the number ofproduction runs and total cost are more influential than that ofsetup learning. The results are also consistent with the important findings ofprevious works in which the forgetting effects are dependent on the learning effects. This paper also provides insight useful to practitioners and researchers in understanding the interactive effects of the five demand patterns and nine relationships generated by learning and forgetting rates on the average optimal number of production runs and the average optimal total cost.

Acknowledgements

Dr. H. M. Chen wishes to thank the National Science Council of R.O.C. for funding this research. The authors thank the anonymous referees and the editor for their constructive and helpful comments on the earlier version ofthis paper.

Appendix A

The derivation ofthe inventory carrying cost incurred in Phase I.

(13)

As shown inFig. 1, the demand of Kj periods (from period i to period i þ Kj 1) is satisﬁed in Phase I.

The inventory level at the beginning ofperiod i is I ði 1Þ: Given that Uði; wÞ ¼ ðPw

a¼i daÞ Iði 1Þ; the units produced and delivered in this phase are di Iði 1Þ; d_iþ1; y; and diþKj1; respectively. Hence, the production cost for each delivery can be given, respectively, as follows:

Cdi ¼ C_o ^dⁱ^Iði1ÞX

x¼1

tj; xþ C_m½d_i Iði 1Þ ¼ C_o ^{U ði;iÞ}X

x¼1

tj; xþ C_mU ði; iÞ;

C_d_iþ1 ¼ C_o ^d^iþ1^þdXⁱ^Iði1Þ

x¼diIði1Þþ1

t_{j; x}þ C_md_iþ1 ¼ C_o ^{U ði;iþ1Þ}X

U ði;iÞþ1

t_{j; x}þ C_md_iþ1; y;

and

CdiþK_j1 ¼ Co

diþK_j1þ?þdXiIði1Þ x¼diþK_j2þ?þdiIði1Þþ1

tj; xþ CmdiþKj1¼ Co

U ði;iþKXj1Þ x¼U ði;iþKj2Þþ1

tj; xþ CmdiþKj1:

The inventory carrying cost incurred in this phase is

WCj¼ C_h ðC_otj;1þ C_mÞðt_j;2þ t_j;3þ ? þ t_j;q_j₁þ t_j;q_jÞ (

þðCotj;2þ CmÞðtj;3þ tj;4þ ? þ tj;q_j1þ tj;q_jÞ þ ? þ ðCotj;q_j2þ CmÞðtj;q_j1þ tj;q_jÞ þ ðCotj;q_j1þ CmÞðtj;q_jÞ

Co

X

U ði;iÞ

x¼1

tj; xþ CmU ði; iÞ

" #

ðBjþ Pj iÞ ^iþKX^j²

w¼i

Co

X

U ði;wþ1Þ

x¼U ði;wÞþ1

tj; xþ Cmdwþ1

" #

½Bjþ Pj ðw þ 1Þ

)

¼ C_h X^q^j1

x¼1

ðC_otj; xþ C_mÞ X^q^j

y¼xþ1

tj; y

!

" #

C_o ^{U ði;iÞ}X

x¼1

tj; xþ C_mU ði; iÞ

" #

ðBjþ Pj iÞ (

^iþKX^j²

w¼i

C_o ^{U ði;wþ1Þ}X

x¼U ði;wÞþ1

t_{j; x}þ C_md_wþ1

!

½B_jþ P_j ðw þ 1Þ

)

: ðA:1Þ

Appendix B

The derivation ofthe inventory carrying cost incurred in Phase II.

Given that the production ofthe jth production run completed in period i þ K_j (i.e., i þ K_j 1pBjþ P_joi þ Kj) and the demands of M_j periods (from period i þ K_j to period i þ K_jþ M_j 1) are satisﬁed in Phase II, as shown in Fig. 1. The d_iþK_j; d_iþK_jþ1; y; and d_iþK_jþMj1 units produced in the jth production run are delivered at the end ofperiod i þ K_j; i þ K_jþ 1; y; and i þ K_jþ M_j 1; respectively. Since Uði; wÞ ¼ Pw

a¼i d_a

Iði 1Þ; the production cost for each delivery can be calculated by

C_d_iþKj ¼ C_o ^d^iþK^j^þ?þdXⁱ^Iði1Þ

x¼diþK_j1þ?þdiIði1Þþ1

t_{j; x}þ C_md_iþK_j ¼ C_o ^{U ði;iþK}X^j^Þ

x¼U ði;iþKj1Þþ1

t_{j; x}þ C_md_iþK_j;

(14)

CdiþKjþ1 ¼ C_o ^d^iþK^j^þ1^þ?þdXⁱ^Iði1Þ

x¼diþK_jþ?þdiIði1Þþ1

tj; xþ C_mdiþKjþ1¼ C_o ^{U ði;iþK}X^j^þ1Þ

x¼U ði;iþKjÞþ1

tj; xþ C_mdiþKjþ1; y;

and

CdiþKjþMj1

¼ C_o ^d^iþK^j^þM^j¹^þ?þdX ⁱ^Iði1Þ

x¼diþK_jþM_j2þ?þdiIði1Þþ1

tj; xþ C_mdiþKjþMj1

¼ C_o ^{U ði;iþK}X^j^þM^j^1Þ

t_{j; x}þ C_md_iþK_j_þM_j₁:

The production cost for the residual units, which are produced in the jth production run but left for the ﬁrst delivery in the next production run, is

CdiþKjþMj ¼ C_o X^q^j

x¼diþK_jþM_j1þ?þdiIði1Þþ1

tj; xþ C_m½Q_j Dð1; i þ K_jþ M_j 1Þ

¼ C_o X^q^j

tj; xþ C_m½Q_j Dð1; i þ K_jþ M_j 1Þ:

As a result, the inventory carrying cost incurred during Phase II can be expressed as HC_j¼ C_h ^iþKX^j^þM^j¹

a¼iþKj

C_o ^{U ði;aÞ}X

x¼U ði;a1Þþ1

t_{j; x}þ C_md_a

!

½a ðB_jþ P_jÞ

8<

:

þ C_o X^q^j

tj; xþ C_mðQ_j Dð1; i þ K_jþ M_j 1ÞÞ 2

4

3

5½i þ Kjþ M_j ðB_jþ P_jÞ

9=

;: ðB:1Þ

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